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f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"f2dc07e53aa7126326ee9eefe8d713f4648db427cc1482c2e326722afea6a4a5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4155b181fd90d601","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     3,         6;\n  2:     9,        96,          27;\n  3:    27,      1002,        1503,          108;\n  4:    81,      8988,       39078,        17976,           405;\n  5:   243,     75630,      750690,      1000920,        189075,          1458;\n  6:   729,    618648,    12368205,     36033120,      20613675,       1855944,          5103;\n  7:  2187,   4997298,   186769881,   1027887780,    1284859725,     373539762,      17490543,       17496;\n  8:  6561,  40144596,  2675374704,  25474722264,   59103547350,   38212083396,    6242540976,   160578384,      59049;\n  9: 19683, 321721014, 37068113124, 576261487728, 2250996868290, 2701196241948, 1008457603524, 98848301664, 1447744563, 196830;\n","recurrence_sentence":"The row polynomials satisfy (5*n-2)*P(n,x) = (30*n+18+(100*n^2-80*n-84)*x-(125*n^3-150*n^2-65*n+42)*x^2)*P(n-2,x) + ((50*n+30)*x-(125*n^2+75*n)*x^2+(125*n^2+25*n-30)*x^3)*d/dx P(n-2,x) + (5*n-12+(50*n^2-40*n+8)*x)*P(n-1,x) + ((25*n-10)*x-(25*n-10)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((30*n + 18 + (100*n^2 - 80*n - 84)*x - (125*n^3 - 150*n^2 - 65*n + 42)*x^2)*P[n - 2, x] + ((50*n + 30)*x - (125*n^2 + 75*n)*x^2 + (125*n^2 + 25*n - 30)*x^3)*D[P[n - 2, x], x] + (5*n - 12 + (50*n^2 - 40*n + 8)*x)*P[n - 1, x] + ((25*n - 10)*x - (25*n - 10)*x^2)*D[P[n - 1, x], x])/(5*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((30*n + 18 + (100*n^2 - 80*n - 84)*x - (125*n^3 - 150*n^2 - 65*n + 42)*x^2)*v[i - 2] + ((50*n + 30)*x - (125*n^2 + 75*n)*x^2 + (125*n^2 + 25*n - 30)*x^3)*deriv(v[i - 2], x) + (5*n - 12 + (50*n^2 - 40*n + 8)*x)*v[i - 1] + ((25*n - 10)*x - (25*n - 10)*x^2)*deriv(v[i - 1], x))/(5*n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4155b181fd90d601","steps":["\u03b8+1"],"edges":["e-e8f50a7eb5d526d1"],"text":"A257623 \u2192 \u03b8+1"},"headline":"A257623 \u2192 \u03b8+1","chain":null}